Problem: The sum of two numbers is $45$, and their difference is $13$. What are the two numbers?
Explanation: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 45}$ ${x-y = 13}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 58 $ $ x = \dfrac{58}{2} $ ${x = 29}$ Now that you know ${x = 29}$ , plug it back into $ {x+y = 45}$ to find $y$ ${(29)}{ + y = 45}$ ${y = 16}$ You can also plug ${x = 29}$ into $ {x-y = 13}$ and get the same answer for $y$ ${(29)}{ - y = 13}$ ${y = 16}$ Therefore, the larger number is $29$, and the smaller number is $16$.